Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
If $\text{y}=\sin(\sin\text{x})$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0$
✓

Answer

Given,
$\text{y} = \sin (\sin \text{x})\dots\text{ eq. } 1$
To prove: $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\tan\text{x}.\frac{\text{dy}}{\text{dx}}+\text{y}\cos^2\text{x}=0$
Let's find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
As, $\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\Big(\frac{\text{dy}}{\text{dx}}\Big)$
So, lets first find $\frac{\text{dy}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\sin(\sin\text{x})$
Using chain rule, we will differentiate the above expression:
Let $\text{t}=\sin\text{x}\Rightarrow\frac{\text{dt}}{\text{dx}}=\cos\text{x}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{dy}}{\text{dy}}\frac{\text{dt}}{\text{dx}}$
$\frac{\text{dy}}{\text{dx}}=\cos\text{t}\cos\text{x}=\cos(\sin\text{x})\cos\text{x}\dots\text{ eq. 2}$
Again differentiating with respect to x applying product rule:
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=\cos\text{x}\frac{\text{d}}{\text{dx}}\cos(\sin\text{x})+\cos(\sin\text{x})\frac{\text{d}}{\text{dx}}\cos\text{x}$
Using chain rule again in the next step:
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\cos\text{x}\cos\text{x}\sin(\sin\text{x})-\sin\text{x}\cos(\sin\text{x})$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\text{y}\cos^2\text{x}-\tan\text{x}\cos\text{x}\cos(\sin\text{x})$
$[$using eq. 1: $\text{y} = \sin (\sin \text{x})]$
And using eq. 2, we have:
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\text{y}\cos^2\text{x}-\tan\text{x}\frac{\text{dy}}{\text{dx}}$
$\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}\cos^2\text{x}+\tan\text{x}\frac{\text{dy}}{\text{dx}}=0\dots\text{ proved.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the vector equations of the coordinate planes.
→
Solve the following for x and y.$\begin{bmatrix}3&-4\\9&2\end{bmatrix}\begin{bmatrix}\text{x}\\\text{y}\end{bmatrix}=\begin{bmatrix}10\\2\end{bmatrix}$
→
Show that the exponential function $f : R \rightarrow R,$ given by $f(x) = e^x,$ is one-one but not onto. What happens if the co-domain is replaced by $R + 0R0 + ($set of all positive real numbers$)?$
→
A ladder 13m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5m/ sec. How fast is the angle $\theta$ between the ladder and the ground is changing when the foot of the ladder is 12m away from the wall.
→
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹  y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.
→
Verify Rolle's theorem for the following function on the indicated intervals$\text{f}(\text{x})=\sin2\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$
→
A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs. 60 per unit for the product A and Rs. 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.
→
Evaluate: $\int\frac{\text{dx}}{\text{x}\text{(x}^{5}\text{+3)}}$
→
Show that $\text{f(x)}=\cos\text{x}^2$ is a continuous function.
→
In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?
→